How do you prove that the order of a in Zn (where n is an integer greater than or equal to 1) viewed as a group of order n with respect to addition is
n/(gcd(a,n))?
Let be a cyclic group (finite). And let . It means the has a generator that is for some .
Let thus, for .
We need to find the smallest such as,
By the properties of cyclic groups and the fact that is a generator it is equivalent to saying divides .
Thus, we need the smallest such that,
is an integer.
We can write it as, (by dividing by
But,
Thus,
divides .
The smallest such is of course,
Thus, the order of any element is,